When \(\mathrm {ord}_{s=1}L(E,s) \le 1\) , the Iwasawa main conjecture is not used to have Theorem REF .
In particular, we obtain an explicit formula for the exact size of \(\textrm {{\normalfont \selectfont SH}}(E/\mathbb {Q})[p^\infty ]\)  for semi-stable elliptic curve \(E\)  of analytic rank \(\le 1\)  and every prime \(p \ge 5\)  with the surjectivity condition but with no use of the Iwasawa main conjecture
(cf. [1]}, [2]}).
When \(E\)  is semi-stable and \(p \ge 11\) , \(\overline{\rho }\)  is always surjective [3]}.
We refer to \(§\)REF  for details.
